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resources also from neighboring cells.Notice that if we include sunlight among
the resources contained by a cell, we can model in this way also the “shade”
effect.Seeds are also introduced in the model as a resource that moves from cell
to cell.Thus, a tree can scatter its seeds in the neighboring cells.
Now, let
C
(
i, j
) be the cell located at position (
i, j
) in the lattice.With
R
(
i, j
)
we will denote the resource vector of cell
C
(
i, j
), with
M
(
i, j
) the maximum
resource values, and so on.The transition function can be divided in four sub-
steps, defined as follows.
Tree sustenance.
If a tree is present in cell
C
(
i, j
), it takes from it a given
quantity (defined by
N
T
(
i, j
)) of each available resource
R
(
i, j
).If, for some
resource
i
, the amount available
r
i
(
i, j
) is lower than the corresponding value in
N
T
(
i, j
), then the tree takes the whole quantity
r
i
(
i, j
).The amount of resources
taken depends on the size of the tree
Z
T
(
i, j
).Then, if enough resources (those
taken at this step, plus the resources stored at previous steps), are available, as
defined by vector
U
T
(
i, j
), the tree grows, that is, each part grows according
to the growth rate vector
G
T
(
i, j
) associated with the tree.Else, the resources
might be just su
"
cient for the tree to survive (vector
U
T
(
i, j
)).In this case, the
tree parameters are left unchanged.In both cases, the tree “burns” an amount of
each resource, as defined by vector
U
T
(
i, j
)or
U
T
(
i, j
).All the unused resources
collected at this step are stored and added to vector
R
T
(
i, j
).Otherwise, if
the overall amount (stored plus collected) of at least one resource is under the
“survival threshold” of the tree, the latter dies.The tree also dies when it reaches
its maximum age defined in vector
M
T
(
i, j
).
Tree reproduction.
We have two cases to consider: a tree is present in the
cell, or the cell is empty.In the former case, the tree may produce some seeds
(if it is old enough, and according to the size of its fruits
z
T
(
i, j
)), that are used
to update the corresponding variable in the seed vector
S
(
i, j
).Also, new trees
cannot sprout from the seeds contained in the cell if a tree is already present.
Instead, the cell can be vacant and contain some seeds.If the resources present
in the cell are su
"
cient (quantities defined as global parameters for each tree
species) a new tree is born.If seeds from different species are present in the cell,
the winning species is chosen at random, with probability proportional to the
number of its seeds.
Resource production.
In the third sub-step, each cell produces a given
amount of resources, according to its production vector
P
(
i, j
).In any case, the
amount of each resource contained in the cell cannot exceed the corresponding
maximum value defined by vector
M
(
i, j
).
Resource flow.
In this step, resources are balanced among neighboring cells,
in order to let resources flow from richer to poorer cells.Let
r
h
(
i, j
)bethe
amount of resource
h
contained by cell
C
(
i, j
), and assume that we are using the
